Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem

Amoura, Karima; Bernardi, Christine; Chorfi, Nejmeddine

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 5, pp. 897-921.

We consider the Stokes problem provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. Next we propose a discretization by spectral element methods which relies on this formulation. A detailed numerical analysis leads to optimal error estimates for the three unknowns and numerical experiments confirm the interest of the discretization.

Classification : 35Q30, 65N35
Mots clés : Stokes problem, vorticity, velocity and pressure formulation, spectral element methods

[1] M. Amara, D. Capatina-Papaghiuc, E. Chacón-Vera and D. Trujillo, Vorticity-velocity-pressure formulation for Navier-Stokes equations. Comput. Vis. Sci. 6 (2004) 47-52.

[2] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Method. Appl. Sci. 21 (1998) 823-864. | Zbl

[3] F. Ben Belgacem and C. Bernardi, Spectral element discretization of the Maxwell equations. Math. Comput. 68 (1999) 1497-1520. | Zbl

[4] C. Bernardi and N. Chorfi, Spectral discretization of the vorticity, velocity and pressure formulation of the Stokes problem. SIAM J. Numer. Anal. 44 (2006) 826-850. bibitemBMx C. Bernardi and Y. Maday, Spectral Methods, in the Handbook of Numerical Analysis V, P.G. Ciarlet and J.-L. Lions Eds., North-Holland (1997) 209-485. | Zbl

[5] C. Bernardi, M. Dauge and Y. Maday, Polynomials in the Sobolev world. Internal Report, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (2003).

[6] C. Bernardi, V. Girault and P.-A. Raviart, Incompressible Viscous Fluids and their Finite Element Discretizations, in preparation.

[7] J. Boland and R. Nicolaides, Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731. | Zbl

[8] A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Method. Appl. Sci. 24 (2001) 31-48. | Zbl

[9] A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101 (2005) 29-65. | Zbl

[10] M. Costabel and M. Dauge, Espaces fonctionnels Maxwell: Les gentils, les méchants et les singularités, Web publication (1998) http://perso.univ-rennes1.fr/monique.dauge.

[11] M. Costabel and M. Dauge, Computation of resonance frequencies for Maxwell equations in non smooth domains, in Topics in Computational Wave Propagation, M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne Eds., Springer (2004) 125-161. | Zbl

[12] F. Dubois, Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091-1119. | Zbl

[13] F. Dubois, M. Salaün and S. Salmon, Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem. J. Math. Pure. Appl. 82 (2003) 1395-1451. | Zbl

[14] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | MR | Zbl

[15] J.-C. Nédélec, Mixed finite elements in ${ℝ}^3$. Numer. Math. 35 (1980) 315-341. | Zbl

[16] P.-A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes Eds., Lect. Notes Math. 606, Springer-Verlag (1977) 292-315. | Zbl

[17] S. Salmon, Développement numérique de la formulation tourbillon-vitesse-pression pour le problème de Stokes. Ph.D. thesis, Université Pierre et Marie Curie, Paris (1999).